Question

Suppose $A,B,C$ are defined as $A=a^{2}b+a{b}^2-a^{2}c-a{c}^2,B=b^{2}c+b{c}^2-a^{2}b-a{b}^2,C=a^{2}c+a{c}^2-b^{2}c-b{c}^2$, where $a>b>c>0$ and the equation $A{x}^2+Bx+C=0$ has equal roots, then $a,b,c$ are in ..................

• A. H.P
• B. A.P
• C. G.P
• D. Both a and c

Answer 1

The correct option is A H.P

Given $A=a^{2}b+{ab}^2-a^{2}c-{ac}^2$

$=a^2(b-c)+a(b^2-c^2)$

$=a(b-c)(a+b+c)$

Similarly $B=b(c-a)(a+b+c),$

$C=c(a-b)(a+b+c)$

Now $A{x}^2+Bx+C=0$

$\Rightarrow (a+b+c)\{a(b-c)x^2+b(c-a)x-(a-b)\}=0$ ...(1)

has equal roots

Clearly $1$ is a root of equation (1)

So, other root is also $1.$

$\Rightarrow$ Product of roots of equation (1) $=1$

$\Rightarrow \frac{c(a-b)}{a(b-c)}=1\Rightarrow b=\frac{2ac}{a+c}$

Hence $a,b,c$ are in $H.P.$